Numbers: Real, Or What?

Our friend Dr. Bill Vallicella, the Maverick Philosopher, has written a post arguing that numbers have an eternal, mind-independent existence as Platonic abstracta. This is of course a respectable and widely held opinion, with an ancient pedigree. I’m leery of it, though: I think numbers are what minds invent to make useful models of certain aspects of the world, and didn’t exist in any way at all until minds came along to deploy them.

Bill writes:

Our question is whether numbers themselves are mental constructions, not whether numerals are mental constructions. This is connected with the question of whether mathematics is in any sense conventional. No doubt notation systems are conventional, i.e. decided upon by human beings (or whatever other intelligent critters there might be elsewhere); but it doesn’t follow that numbers or other mathematical objects are.

If numbers themselves are mental constructions, then they depend on our existence for their existence. Their existence is a mental existnce in or before our minds, and thus a dependent mode of existence. (Forget about extraterrestrial intelligences for the nonce.) The same goes for the truths in which they are involved. (Thus 7 and 9 and 16 are involved in the truth expressed by ‘7 + 9 = 16’.) But we didn’t always exist. So if numbers depend ion us, they they didn’t always exist. Consider a time before any minds existed, some time after the Big Bang and before the emergence of life on earth, say.

During that interval, the speed of light and the speed of sound were the same as they are now, and during that time the former was greater than the latter, as is the case now. Let ‘c’ denote the speed of light in a vacuum. C is identical to some number, which number depending on the units of measurement one employs. So c = 186,000 miles/sec (approximately). In the metric system, c = 300,000 km/sec (approximately). The point is that once the system of measurement is fixed — which of course is conventional — then some definite number is the SOL. Similarly with the speed of sound, SOS. Now

1. SOL > SOS

is true now and was true at the time when no humans existed. Of course, at that time the concept or notion or idea greater than (taken in its mathematical sense) did not exist since concepts (notions, ideas) cannot exist except ‘in’ a mind. (‘In’ here not to be taken spatially.) But the mathematical relation picked out by ‘>’ existed.

For if it did not, then (1) could not have been true at the time in question. And the same goes for the relational fact of SOL’s being greater than SOS. That fact obtained at the time when no minds existed. So its constituents (the numbers and the greater than relation) had to exist at that time as well.

Therefore, mathematical objects cannot be our mental creations.

Hmmm. My question for Bill is:

Why is “speed” itself anything more than human shorthand for modeling a particular aspect of the different configurations of the world at different times? Let’s say that spacetime event A is “a car over here at some instant of time”, and event B is “the same car over there at some other instant of time”. It’s not until we come along, with a need to model this in a handy way, that we develop the concept of “speed” as

(the spatial component of the separation of events A and B, represented by our minds as a number)

…divided by…

(the temporal component of the separation of events A and B, represented by our minds as a number).

But in a world without minds to make such representations, there’s nothing intrinsically numerical about a separation; it’s just an event over here, and another over there. It’s not until we seek to quantify it, for our own purposes, that numbers enter the picture.

I think, then, that Bill is smuggling in his numbers when he introduces the concept of “speed”, and pulling them out again at the end.

(I realize there probably wouldn’t be a car in the first place in a world without minds, but you get the point: the world just IS. It isn’t until we needed to describe it that we needed numbers, so we invented them.)

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26 Comments

  1. Tiptoeing cautiously into this mental minefield, are we not discussing the old ‘unseen, unheard tree falling in a forest therefore it never happened’ problem? By implying that certain matters only exist when human minds are there to, as it were, ‘invent’ them, we are in danger of denying the existence of anything before human life.

    Also, I think (tentatively) that there is a difference between numbers and mathematics. At some time, around where I live, let us suppose that there was a herd of 50 dinasours in an area of say 100 square miles. That was surely ‘true’ even if no-one, let alone a mathematician, was around to count and to measure.

    What I’m trying to say, rather badly, is that there were many things whose existence we can infer from our human knowledge but they would still have existed even if all human life is extinguished in the next 5 minutes! (Crikey, what was that?)

    And (he went on, tediously) the very nanosecond that something, anything, comes into existense then quantity (numbers) exists, too, even if there is no-one to comprehend it.

    Posted September 16, 2011 at 5:34 am | Permalink
  2. bob koepp says

    Malcolm –
    You think that “numbers are what minds invent to make useful models of certain aspects of the world, and didn’t exist in any way at all until minds came along to deploy them.” It seems to me that the “aspects of the world” to which you refer are irreducibly quantitative properties and relations. A great many philosophers and physicists have tried, unsuccessfully, to reconstruct physical theory without making “essential reference” to such quantitative properties and relations. I’m not just saying that the models employ numbers. Rather, it appears that successful (i.e., empirically adequate) models employ numbers because the world being modeled is “numbered.”

    Posted September 16, 2011 at 8:36 am | Permalink
  3. Malcolm says

    David, I’d respond by saying that there may be a bunch of dinosaurs out there in the world, but that it is only we humans who have come up with the idea of numbering them.

    When you say:

    What I’m trying to say, rather badly, is that there were many things whose existence we can infer from our human knowledge but they would still have existed even if all human life is extinguished in the next 5 minutes! (Crikey, what was that?)

    …you’re just asserting the very proposition that’s in dispute here.

    Posted September 16, 2011 at 12:29 pm | Permalink
  4. Malcolm says

    Bob,

    You wrote:

    A great many philosophers and physicists have tried, unsuccessfully, to reconstruct physical theory without making “essential reference” to such quantitative properties and relations.

    But that’s the point: numbers are what our human minds want, and so create, for our models (theories) of the world. But I don’t see it as self-evident that prior to the arrival of philosophers and theorists that the numbers themselves were “out there” in the world all along.

    Posted September 16, 2011 at 12:34 pm | Permalink
  5. Malcolm, I grant that what I said is an assertion, as is what you said! However, I think mine has some slight authority in that dinasours *did* exist, irrespective of whether humans knew about them or not. Thus, my contention that numbers also existed leans on that *logical* inference. Or to put it more bluntly, if there were fifty of them there were fifty of them, irrespective of whether anyone was around to count them!

    Butting, rather foolishly, into someone else’s argument, I take your point that numbers are a requirement for formulating models of the world, but so were stones required in formulating the building of shelters which gradually turned into palaces.

    Posted September 16, 2011 at 1:54 pm | Permalink
  6. Malcolm says

    I take your point that numbers are a requirement for formulating models of the world, but so were stones required in formulating the building of shelters which gradually turned into palaces.

    Right, David. And just as models of the world are mental “palaces” that did not exist until there were minds to build them, so, I think, are the numbers of which they are constructed.

    Posted September 16, 2011 at 2:26 pm | Permalink
  7. bob koepp says

    Malcolm –
    Philosophers and physicists have also been unable to dispense with essential reference to other sorts of things besides quantities/numbers. Since you’re pushing an extreme sort of nominalism about quantities, why not just swallow the whole pig and claim that since nobody needs to refer to the world until we want to describe some feature(s) of it, the world doesn’t exist apart from our descriptions? You’d be in respectable (if not “good”) company.

    Posted September 16, 2011 at 2:56 pm | Permalink
  8. “And just as models of the world are mental “palaces” that did not exist until there were minds to build them, so, I think, are the numbers of which they are constructed.”

    But the stones to which I referred *did* exist before man came along and built things with them although there was no-one around to call them ‘stones’. Equally, quantities and distances existed before man came along and the only thing we added was names for them, followed by more or less complicated ‘models’.

    I’m off to bed now, and I’m going to put my brain in the glass of water with my teeth because it hurts!

    Posted September 16, 2011 at 3:08 pm | Permalink
  9. Malcolm says

    Bob:

    …why not just swallow the whole pig and claim that since nobody needs to refer to the world until we want to describe some feature(s) of it, the world doesn’t exist apart from our descriptions?

    David:

    But the stones to which I referred *did* exist before man came along and built things with them although there was no-one around to call them ‘stones’.

    Both of you are conflating two classes of objects here, I think. I’m only questioning the existence of Platonic abstracta, not stones and the rest of the physical world.

    Posted September 16, 2011 at 3:24 pm | Permalink
  10. Damn! Nearly droopped my teeth!

    Look, numbers are not ‘abstracta’ they are are real and tangible. With the possible exception of infinity, they are attached to real things in the same way that colour and other descriptive or prescriptive characteristics are attached to things.

    I grant you (hesitantly because I may retract in the morning!) that some of the constructions carried out by man might lead to fanciful conclusions which only exist because of man, but that is *mathematics* and what I am talking of are *numbers* which, in my opinion, pre-exist man. I suppose a simple illustration is to say that it is the difference between arithmetic and algebra.

    Posted September 16, 2011 at 3:43 pm | Permalink
  11. Malcolm says

    Hi David,

    Look, numbers are not ‘abstracta’ they are are real and tangible.

    “Tangible”! Well, with that you’re staking out a pretty radical position; what I’m taking on in this post is the idea that numbers exist as Platonic abstracta as opposed to having no independent existence at all.

    I think it’s hard to argue that numbers are “real and tangible” in the same way that, say, stones are; after all, I can stub my toe on a stone, but not on the number five. And if you put five stones in my hand, what’s tangible is just the stones, not the “five”. I can count them, and say “there are five stones here”, but that’s just a mental operation in which I quantify the world in the way that human minds are prone to do. What’s actually out there in the world, though, is just the stones.

    Posted September 16, 2011 at 4:32 pm | Permalink
  12. Jesse Kaplan says

    This is a hoary topic, but I’ve never seen it explicitly linked to trees-falling-in-forests, which I find rather acute. As I take the opposite view, to me Malcolm seems in effect really to confuse numbers and numerals. To me, numerals are just sort of the material world extrusion of the mental concept of number, which in turn is the human representation of the Platonic abstraca. I was leery of getting drawn into a debate I’ve had innumerable times, but I did have the idea that in a world of quantum potentia, perhaps it really does take a human mind measuring them to make quantifiable discrete entities “collapse” out of a non-discrete soup. I thought of this as a charitable way to redeem Malcolm, but of course in my world that soup is an alphabet soup that includes letters and numbers:)

    Posted September 16, 2011 at 4:51 pm | Permalink
  13. Malcolm says

    No, no, of course I’m not confusing numbers and numerals. Did I really need to take pains to clarify that? (Bill did so, if you read his post, but I didn’t bother to excerpt that part.)

    Posted September 16, 2011 at 5:04 pm | Permalink
  14. Jesse Kaplan says

    entities’

    Posted September 16, 2011 at 5:54 pm | Permalink
  15. bob koepp says

    Malcolm –

    It won’t do to claim that you are simply questioning a Platonic view of numbers. Not when you make such statements as:

    “But in a world without minds to make such representations, there’s nothing intrinsically numerical about a separation; it’s just an event over here, and another over there. It’s not until we seek to quantify it, for our own purposes, that numbers enter the picture.”

    Posted September 16, 2011 at 6:19 pm | Permalink
  16. Malcolm says

    I don’t follow you, Bob. Questioning a Platonic view of numbers is exactly what I’m doing.

    How does what I said contradict that?

    Posted September 16, 2011 at 7:12 pm | Permalink
  17. bob koepp says

    There are varieties of realism about quantities/numbers that are not “Platonic.” For example, abstracta might be viewed as “immanent” in concreta. Unlike Platonic abstracta, they wouldn’t exist independently of the concrete objects in which they are instantiated; but they would be just as mind-independent as those concrete objects. I think that’s how most of the philosphers and physicists to whom I referred earlier would view matters.

    Posted September 16, 2011 at 8:00 pm | Permalink
  18. Malcolm says

    The alternative to Platonism you describe is certainly not the one Bill was talking about, and that I was responding to.

    Bill’s argument is that numbers are Platonic abstracta, and I have been arguing that I remain unconvinced.

    Frankly, I find the “immanent in concreta” middle-ground you’re suggesting here least appealing of all, because it doesn’t account for, say, number theory, or anything else that doesn’t involve concrete objects.

    Posted September 16, 2011 at 8:15 pm | Permalink
  19. Jesse Kaplan says

    I guess the apostrophe is optional. Bob picked up what gave me my bright, rehabilitative idea.

    Of course the confusion isn’t exactly between numbers and numerals, but to those of us on the other side it looks like the same sort of blind spot.

    Posted September 16, 2011 at 10:02 pm | Permalink
  20. bob koepp says

    Malcolm –
    These are difficult topics, and they need to be approached carefully and deliberately.

    It is true that Bill did refer to “abstract or ideal or Platonic objects,” as a plausible alternative to the view that numbers must be either physical objects or mental constructions. But his argument, which you reproduced above, is not a brief for Platonism as such, being directed simply against the view that numbers are mental constructions. The positive side of the argument he presents is based entirely on the notion that the world contained quantitative properties and relations long before there were any minds to do any “constructing.” That argument does not presuppose Platonism.

    Also, you might want to examine more carefully the “immanent in concreta” view (probably first articulated by Aristotle). Why would it not be able to account for the role of numbers in number theory? Contrary to what you seem to suggest, the idea that numbers are immanent in concreta doesn’t obviously imply that the relations between numbers are relations between concrete objects.

    Posted September 17, 2011 at 8:01 am | Permalink
  21. bob koepp says

    Further clarification is probably advisable. It is common in philosophy of mathematics to use ‘Platonism’ and ‘realism’ as synonyms. To that extent, Malcolm is, indeed, objecting to mathematical Platonism. But I don’t think this should be an argument about labels, which is why I mentioned forms of realism that are not “full-bodied” Platonism.

    Posted September 17, 2011 at 9:46 am | Permalink
  22. Malcolm says

    Well, for clarity’s sake, then, if you like: My only point in this post is to question Bill’s argument that numbers cannot be mental objects.

    Posted September 17, 2011 at 11:28 am | Permalink
  23. bob koepp says

    And.. it isn’t clear to me that you have actually responded to the positive argument he presents. In what sense was the world “devoid” of quantitative properties and relations prior to our devising theories about that world?

    Posted September 17, 2011 at 11:57 am | Permalink
  24. Malcolm says

    No, Bob, the onus probandi is on Bill, because he claims to have demonstrated that numbers must have a mind-independent existence.

    All I am saying is that he hasn’t done so. I’m not saying that the natural world is devoid of properties that can be modeled using numbers should a mind choose to do so; I’m saying that Bill hasn’t demonstrated that the numbers themselves necessarily exist simply because such numerically describable properties exist.

    Posted September 17, 2011 at 1:11 pm | Permalink
  25. bob koepp says

    Malcolm –
    I’m sorry to be such a pain about this, but Bill is usually very careful not to claim to provide knock-down demonstrative proofs. In the present case he only claimed that it’s plausible to view numbers as “abstract or ideal or Platonic objects, … ”once one considers the difficulties with the view that numbers are mental constructions.” And the sort of difficulty that he presents for our consideration has been thought persuasive by some pretty smart people, like Quine and Putnam.

    Posted September 17, 2011 at 3:48 pm | Permalink
  26. Malcolm says

    Bob,

    At the risk of being stubborn myself, I must point out that Bill ended his post with:

    “Therefore, mathematical objects cannot be our mental creations.”

    Pretty clear, I think, and my point is that I don’t think he has made his case.

    I’m not unfamiliar with the range of opinions on this topic. I remain unconvinced that Bill’s assertion is true.

    Posted September 17, 2011 at 9:25 pm | Permalink